scholarly journals Paracontact Metric κ , μ -Manifold Satisfying the Miao-Tam Equation

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Dehe Li ◽  
Jiabin Yin
Keyword(s):  
Constant Curvature ◽  
Dimensional Manifold ◽  
Critical Equation ◽  
Negative Constant

In this paper, we classified the paracontact metric κ , μ -manifold satisfying the Miao-Tam critical equation with κ > − 1 . We proved that it is locally isometric to the product of a flat n + 1 -dimensional manifold and an n -dimensional manifold of negative constant curvature − 4 .

scholarly journals Ricci almost solitons and gradient Ricci almost solitons in $(k,\mu)$-paracontact geometry

2017 ◽  
Vol 37 (3) ◽  
pp. 119 ◽  
Author(s):  
Uday Chand De ◽  
Krishanu Mandal
Keyword(s):  
Vector Field ◽  
Constant Curvature ◽  
Einstein Manifold ◽  
Dimensional Manifold ◽  
Reeb Vector ◽  
Potential Vector ◽  
Reeb Vector Field ◽  
Negative Constant ◽  
Potential Vector Field

The purpose of this paper is to study Ricci almost soliton and gradient Ricci almost soliton in $(k,\mu)$-paracontact metric manifolds. We prove the non-existence of Ricci almost soliton in a $(k,\mu)$-paracontact metric manifold $M$ with $k<-1$ or $k>-1$ and whose potential vector field is the Reeb vector field $\xi$. Further, if the metric $g$ of a $(k,\mu)$-paracontact metric manifold $M^{2n+1}$ with $k\neq-1$ is a gradient Ricci almost soliton, then we prove either the manifold is locally isometric to a product of a flat $(n+1)$-dimensional manifold and an $n$-dimensional manifold of negative constant curvature equal to $-4$, or, $M^{2n+1}$ is an Einstein manifold.

TOPONOGOV-TYPE AREA COMPARISON THEOREM OF 2-DIMENSIONAL MANIFOLDS

2013 ◽  
Vol 15 (03) ◽  
pp. 1350007
Author(s):  
XIAOLE SU ◽  
HONGWEI SUN ◽  
YUSHENG WANG
Keyword(s):  
Riemannian Manifold ◽  
Comparison Theorem ◽  
Constant Curvature ◽  
Dimensional Manifold ◽  
Geodesic Triangle ◽  
Type Area ◽  
Simply Connected

Let △p1p2p3 be a geodesic triangle on M, a complete 2-dimensional Riemannian manifold of curvature ≥ k, and let [Formula: see text] be its comparison triangle on [Formula: see text] (a complete and simply connected 2-dimensional manifold of constant curvature k). Our main result is that if △p1p2p3 is areable, then its area is not less than that of [Formula: see text].

scholarly journals GENERALIZATION OF HASIMOTO'S TRANSFORMATION

2009 ◽  
Vol 06 (04) ◽  
pp. 625-630 ◽  
Author(s):  
MATHIEU MOLITOR
Keyword(s):  
Schrödinger Equation ◽  
Nonlinear Schrödinger Equation ◽  
Constant Curvature ◽  
Schrodinger Equation ◽  
Three Dimensional ◽  
Nonlinear Schrodinger Equation ◽  
Vortex Filament ◽  
Dimensional Manifold ◽  
Moving Frames ◽  
Natural Complex

In this paper, we generalize the famous Hasimoto's transformation by showing that the dynamics of a closed unidimensional vortex filament embedded in a three-dimensional manifold M of constant curvature, gives rise under Hasimoto's transformation to the nonlinear Schrödinger equation. We also give a natural interpretation of the function ψ introduced by Hasimoto in terms of moving frames associated to a natural complex bundle over the filament.

scholarly journals HYDROGEN ATOM AS AN EIGENVALUE PROBLEM IN 3-D SPACES OF CONSTANT CURVATURE AND MINIMAL LENGTH

1999 ◽  
Vol 14 (35) ◽  
pp. 2463-2469 ◽  
Author(s):  
L. M. NIETO ◽  
M. SANTANDER ◽  
H. C. ROSU
Keyword(s):  
Wave Function ◽  
Hydrogen Atom ◽  
Constant Curvature ◽  
Three Dimensional ◽  
Minimal Length ◽  
Positive Zero ◽  
Spaces Of Constant Curvature ◽  
Curvature Effects ◽  
Negative Constant ◽  
Space Curvature

An old result of Stevenson [Phys. Rev.59, 842 (1941)] concerning the Kepler–Coulomb quantum problem on the three-dimensional (3-D) hypersphere is considered from the perspective of the radial Schrödinger equations on 3-D spaces of any (either positive, zero or negative) constant curvature. Further to Stevenson, we show in detail how to get the hypergeometric wave function for the hydrogen atom case. Finally, we make a comparison between the "space curvature" effects and minimal length effects for the hydrogen spectrum.

Non-Euclidean Extensions

2018 ◽  
Author(s):  
Geoffrey Hellman ◽  
Stewart Shapiro
Keyword(s):  
Hyperbolic Space ◽  
Constant Curvature ◽  
Positive Curvature ◽  
Spherical Geometry ◽  
Two Dimensional ◽  
Actual Infinity ◽  
Constant Positive Curvature ◽  
Negative Constant ◽  
The Given ◽  
Line P

This chapter adapts the foregoing results to present two non-Euclidean theories, both in line with the (semi-)Aristotelian theme of rejecting points, as parts of regions (but working with actual infinity). The first theory is a two-dimensional hyperbolic space, that is, one that has a negative constant curvature. The second theory captures a space of constant positive curvature, a two-dimensional spherical geometry. The task here is to formulate axioms on regions which allow us to prove that (i) there are no infinitesimal regions and (ii) that there are no parallels to any given “line” through any “point” not on the given “line”.

scholarly journals Casimir effect for gauge fields in spaces with negative constant curvature

2005 ◽  
Vol 39 (2) ◽  
pp. 249-252
Author(s):  
A. A. Bytsenko ◽  
M. E. X. Guimarães ◽  
V. S. Mendes
Keyword(s):  
Constant Curvature ◽  
Casimir Effect ◽  
Gauge Fields ◽  
Negative Constant

scholarly journals CARTAN SPACES AND NATURAL FOLIATIONS ON THE COTANGENT BUNDLE

2013 ◽  
Vol 10 (03) ◽  
pp. 1250089 ◽  
Author(s):  
H. ATTARCHI ◽  
M. M. REZAII
Keyword(s):  
Constant Curvature ◽  
Cotangent Bundle ◽  
Negative Constant

In this paper, the natural foliations in cotangent bundle T*M of Cartan space (M, K) is studied. It is shown that geometry of these foliations are closely related to the geometry of the Cartan space (M, K) itself. This approach is used to obtain new characterizations of Cartan spaces with negative constant curvature.

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